Modeling left-truncated and right-censored survival data with longitudinal covariates

There is a surge in medical follow-up studies that include longitudinal covariates in the modeling of survival data. So far, the focus has been largely on right-censored survival data. We consider survival data that are subject to both left truncation and right censoring. Left truncation is well known to produce biased sample. The sampling bias issue has been resolved in the literature for the case which involves baseline or time-varying covariates that are observable. The problem remains open, however, for the important case where longitudinal covariates are present in survival models. A joint likelihood approach has been shown in the literature to provide an effective way to overcome those difficulties for right-censored data, but this approach faces substantial additional challenges in the presence of left truncation. Here we thus propose an alternative likelihood to overcome these difficulties and show that the regression coefficient in the survival component can be estimated unbiasedly and efficiently. Issues about the bias for the longitudinal component are discussed. The new approach is illustrated numerically through simulations and data from a multi-center AIDS cohort study.Comment: Published in at http://dx.doi.org/10.1214/12-AOS996 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Inverse regression for longitudinal data

Sliced inverse regression (Duan and Li [Ann. Statist. 19 (1991) 505-530], Li [J. Amer. Statist. Assoc. 86 (1991) 316-342]) is an appealing dimension reduction method for regression models with multivariate covariates. It has been extended by Ferr\'{e} and Yao [Statistics 37 (2003) 475-488, Statist. Sinica 15 (2005) 665-683] and Hsing and Ren [Ann. Statist. 37 (2009) 726-755] to functional covariates where the whole trajectories of random functional covariates are completely observed. The focus of this paper is to develop sliced inverse regression for intermittently and sparsely measured longitudinal covariates. We develop asymptotic theory for the new procedure and show, under some regularity conditions, that the estimated directions attain the optimal rate of convergence. Simulation studies and data analysis are also provided to demonstrate the performance of our method.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1193 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org). With Correction

Mixed principal eigenvalues in dimension one

This is one of a series of papers exploring the stability speed of one-dimensional stochastic processes. The present paper emphasizes on the principal eigenvalues of elliptic operators. The eigenvalue is just the best constant in the $L^{2}$-Poincar\'e inequality and describes the decay rate of the corresponding diffusion process. We present some variational formulas for the mixed principal eigenvalues of the operators. As applications of these formulas, we obtain case by case explicit estimates, a criterion for positivity, and an approximating procedure for the eigenvalue.Comment: 45 pages; Front. Math. China, 201